A225147 -id:A225147 - cp's OEIS frontend

A349264 Generalized Euler numbers, a(n) = n![x^n](sec(4x)(sin(4x) + 1)).

1, 4, 16, 128, 1280, 16384, 249856, 4456448, 90767360, 2080374784, 52975108096, 1483911200768, 45344872202240, 1501108249821184, 53515555843342336, 2044143848640217088, 83285910482761809920, 3605459138582973251584, 165262072909347030040576, 7995891855149741436305408

Offset: 0

Peter Luschny, Nov 20 2021

nonn

			Exponential generating functions of generalized Euler numbers in context:
egf1 = sec(1*x)*(sin(x) + 1).
   [A000111, A000364, A000182]
egf2 = sec(2*x)*(sin(x) + cos(x)).
   [A001586, A000281, A000464]
egf3 = sec(3*x)*(sin(2*x) + cos(x)).
   [A007289, A000436, A000191]
egf4 = sec(4*x)*(sin(4*x) + 1).
   [A349264, A000490, A000318]
egf5 = sec(5*x)*(sin(x) + sin(3*x) + cos(2*x) + cos(4*x)).
   [A349265, A000187, A000320]
egf6 = sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)).
   [A001587, A000192, A000411]
egf7 = sec(7*x)*(-sin(2*x) + sin(4*x) + sin(6*x) + cos(x) + cos(3*x) - cos(5*x)).
   [A349266, A064068, A064072]
egf8 = sec(8*x)*2*(sin(4*x) + cos(4*x)).
   [A349267, A064069, A064073]
egf9 = sec(9*x)*(4*sin(3*x) + 2)*cos(3*x)^2.
   [A349268, A064070, A064074]

Matthew House, Table of n, a(n) for n = 0..389
William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia, The generating function for the Dirichlet series Lm(s), Mathematics of Computation, Vol. 81, No. 278, pp. 1005-1023, April 2012.
Ruth Lawrence and Don Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999), no. 1, 93-107.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigendum: Generalized Euler and class numbers, Math. Comp. 22, (1968) 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Row 4 of A349271.
Cf. A000111, A001586, A007289, A349265, A001587, A349266, A349267, A349268.
Secant  bisections: A000364, A000281, A000436, A000490, A000187, A000192, A064068, A064069, A064070.
Tangent bisections: A000182, A000464, A000191, A000318, A000320, A000411, A064072, A064073, A064074.
Cf. A225147, A349429.

sec(4*x)*(sin(4*x) + 1): series(%, x, 20): seq(n!*coeff(%, x, n), n = 0..19);

m = 19; CoefficientList[Series[Sec[4*x] * (Sin[4*x] + 1), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 20 2021 *)

seq(n)={my(x='x + O('x^(n+1))); Vec(serlaplace((sin(4*x) + 1)/cos(4*x)))} \\ Andrew Howroyd, Nov 20 2021

A001587 Generalized Euler numbers.

Original entry on oeis.org

2, 6, 46, 522, 7970, 152166, 3487246, 93241002, 2849229890, 97949265606, 3741386059246, 157201459863882, 7205584123783010, 357802951084619046, 19133892392367261646, 1096291279711115037162, 67000387673723462963330, 4350684698032741048452486, 299131045427247559446422446

Offset: 0

N. J. A. Sloane

nonn

These numbers are related to the values at negative integers of the L-functions for two primitive Dirichlet characters of conductor 24. - F. Chapoton, Oct 05 2020

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Lars Blomberg, Table of n, a(n) for n = 0..199
LMFDB, character 24.5
LMFDB, character 24.11
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigendum: Generalized Euler and class numbers, Math. Comp. 22, (1968) 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Bisections are A000192 and A000411. Overview in A349264.

Similar sequences: A000111, A225147.

egf := sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)): ser := series(egf, x, 20): seq(n!*coeff(ser, x, n), n = 0..17); # Peter Luschny, Nov 21 2021

t = PowerSeriesRing(QQ, 't').gen()
f = 2 * (sin(3 * t) + cos(3 * t)) / (2 * cos(4 * t) - 1)
f.egf_to_ogf().list() # F. Chapoton, Oct 06 2020

E.g.f.: 2 (sin(3 x) + cos(3 x)) / (2 cos(4 x) - 1). - F. Chapoton, Oct 06 2020
a(n) ~ 2^(2*n + 2) * 3^(n + 1/2) * n^(n + 1/2) / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Nov 05 2021
a(n) = n!*[x^n](sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x))). - Peter Luschny, Nov 21 2021

a(11)-a(14) from Lars Blomberg, Sep 10 2015

A377666 Array read by ascending antidiagonals: A(n, k) = Sum_{j = 0..k} binomial(k, j) * Euler(j, 0) (2n)^j.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -3, 0, 1, 1, -3, -5, 11, 5, 1, 1, -4, -7, 46, 57, 0, 1, 1, -5, -9, 117, 205, -361, -61, 1, 1, -6, -11, 236, 497, -3362, -2763, 0, 1, 1, -7, -13, 415, 981, -15123, -22265, 24611, 1385, 1

Offset: 0

Peter Luschny, Nov 05 2024

			Array A(n, k) starts:
  [0]  1,  1,   1,   1,    1,       1,        1, ...  A000012
  [1]  1,  0,  -1,   0,    5,       0,      -61, ...  A122045
  [2]  1, -1,  -3,  11,   57,    -361,    -2763, ...  A212435
  [3]  1, -2,  -5,  46,  205,   -3362,   -22265, ...  A225147
  [4]  1, -3,  -7, 117,  497,  -15123,   -95767, ...  A156201
  [5]  1, -4,  -9, 236,  981,  -47524,  -295029, ...  A377665
  [6]  1, -5, -11, 415, 1705, -120125,  -737891, ...
  [7]  1, -6, -13, 666, 2717, -262086, -1599793, ...

Peter Luschny, Generalized Eulerian polynomials. (See last row of the table.)

Rows: A000012, A122045, A212435, A225147, A156201, A377665.

Cf. A377663 (column 3), A377664 (main diagonal), A363393 (column polynomials).

GHZeta := (k, n, m) -> m^(k+1)*Zeta(0, -k, 1/(m*n)):
A := (n, k) -> ifelse(n = 0, 1, n^k*(GHZeta(k, n, 4) - GHZeta(k, n, 2))):
for n from 0 to 7 do lprint(seq(A(n, k), k = 0..7)) od;
# Alternative:
P := proc(n, k) local j; 2*I*(1 + add(binomial(k, j)*polylog(-j, I)*n^j, j = 0..k)) end:
A := n -> Im(P(n, k)): seq(lprint(seq(A(n, k), k = 0..7)), n = 0..7);
# Computing the transpose using polynomials P from A363393.
P := n -> add(binomial(n + 1, j)*bernoulli(j, 1)*(4^j - 2^j)*x^(j-1), j = 0..n+1)/(n + 1):
Column := (k, n) -> subs(x = -n, P(k)):
for k from 0 to 6 do seq(Column(k, n), n = 0..9) od;
# According to the definition:
A := (n, k) -> local j; add(binomial(k, j)*euler(j, 0)*(2*n)^j, j = 0..k):
seq(lprint(seq(A(n, k), k = 0..6)), n = 0..7);

A[n_, k_] := n^k (4^(k+1) HurwitzZeta[-k, 1/(4n)] - 2^(k + 1) HurwitzZeta[-k, 1/(2n)]);

from mpmath import *
mp.dps = 32; mp.pretty = True
def T(n, k):
    p = 2*I*(1+sum(binomial(k, j)*polylog(-j, I)*n^j for j in range(k+1)))
    return int(imag(p))
for n in range(8): print([T(n, k) for k in range(7)])

A(n, k) = n^k*(GHZeta(k, n, 4) - GHZeta(k, n, 2)) where GHZeta(k, n, m) = m^(k+1) * HurwitzZeta(-k, 1/(m*n)) for n > 0, and T(0, k) = 1.
A(n, k) = Im(P(n, k)) where P(n, k) = 2*i*(1 + Sum_{j=0..k} binomial(k, j)*polylog(-j, i)*n^j).
A(n, k) = substitute(x = -n, P(k, x)) where P(n, x) = (1/(n + 1)) * Sum_{j=0..n+1} binomial(n + 1, j) * Bernoulli(j, 1) * (4^j - 2^j)*x^(j-1).

cp's OEIS Frontend

A349264 Generalized Euler numbers, a(n) = n![x^n](sec(4x)(sin(4x) + 1)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A001587 Generalized Euler numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Sage

Formula

Extensions

A377666 Array read by ascending antidiagonals: A(n, k) = Sum_{j = 0..k} binomial(k, j) * Euler(j, 0) (2n)^j.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

cp's OEIS Frontend

A349264 Generalized Euler numbers, a(n) = n!*[x^n](sec(4*x)*(sin(4*x) + 1)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A001587 Generalized Euler numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Sage

Formula

Extensions

A377666 Array read by ascending antidiagonals: A(n, k) = Sum_{j = 0..k} binomial(k, j) * Euler(j, 0) *(2*n)^j.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A349264 Generalized Euler numbers, a(n) = n![x^n](sec(4x)(sin(4x) + 1)).

A377666 Array read by ascending antidiagonals: A(n, k) = Sum_{j = 0..k} binomial(k, j) * Euler(j, 0) (2n)^j.